Bill OReillys position at Fox News grew increasingly tenuous on Tuesday as support from the Murdoch family showed signs of eroding, according to three people briefed on discussions about his future.

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Mr. O’Reillys fate at the network is expected to be discussed on Thursday at a board meeting for Fox Newss parent company, 21st Century Fox. Chief among the considerations is a continuing investigation into Mr. OReillys behavior conducted by the law firm Paul, Weiss, Rifkind, Wharton & Garrison.

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Pressure on the company increased on Tuesday when another woman reported sexual harassment allegations against Mr. OReilly to 21st Century Fox.

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The company recently renewed Mr. OReillys contract, and the Murdoch family, which controls the company, has stood by him as allegations piled up. But after a New York Times article this month detailed harassment allegations against him, more than 50 companies have pulled their advertising from his show.

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Women inside the company have questioned whether the company was serious about fixing a workplace culture beset by repeated allegations of sexual harassment. And calls for his dismissal grew louder Tuesday.

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Mr. OReilly, the top-rated host in cable news, has been on vacation in Italy for the past week and plans to return to his show on Monday. He has received no indication that the company is planning to take him off the air, according to a person familiar with the matter.

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A decision on Mr. OReillys future could come in the next few days.

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If he is forced to depart, it would be a major blow to the network, which in the past year has also lost its founding chairman, Roger E. Ailes, and another prime-time star, Megyn Kelly. It would also be a stunning downfall for Mr. OReilly, who has dominated the prime-time cable news landscape for most of two decades with a pugnacious, anti-political-correctness commentary that appeals to conservative viewers.

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Although Mr. OReilly, 67, brings immense value to 21st Century Fox as a ratings draw and a revenue generator, the Murdoch family will have to weigh those considerations against its other ambitions. Those include overhauling a Fox News culture that has been damaged by repeated sexual harassment allegations and pursuing the acquisition of the British satellite company Sky, long coveted by the family patriarch, Rupert Murdoch. The company must convince British regulators next month that it is fit to acquire the 61 percent in Sky that it does not own.

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Questions about Mr. OReillys future come just weeks after The Times reported that Mr. OReilly or the company had made payouts to five women involving allegations of inappropriate behavior by Mr. OReilly. In exchange, the women agreed not to sue or speak of the allegations. The payouts totaled about $13 million.

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Two of the deals were struck by 21st Century Fox last year after Mr. Ailes was pushed out following accusations by several women of sexual harassment. In the aftermath of Mr. Ailess dismissal, the company said behavior that disrespects women or contributes to an uncomfortable work environment would not be tolerated. The Times reported that two other women who had not received payouts had accused Mr. OReilly of harassing them. One of those women has since reported her accusations to 21st Century Fox.